UPDATE #4:

U.S. Secretary of State Mike Pompeo has spoken with his Japanese and South Korean counterparts after President Donald Trump's summit with North Korean leader Kim Jong Un.

Pompeo tweeted shortly after the summit ended Tuesday that he'd telephoned South Korean Foreign Minister Kang Kyung-wha and Japanese Foreign Minister Taro Kono. The State Department released a photograph of Pompeo on the phone.

Pompeo says on Twitter he provided the diplomats with "a brief readout of today's meeting" between Trump and Kim. The two leaders concluded their summit by signing a document in which Trump pledged "security guarantees" to the North and Kim reiterated his commitment to "complete denuclearization of the Korean Peninsula."

The State Department is declining to release any additional information about the calls.

UPDATE #3:

President Donald Trump and North Korean leader Kim Jong Un have signed a joint document in which they commit to working "toward complete denuclearization of the Korean Peninsula."

The document signed by the leaders at their historic summit Tuesday also says they will join efforts "to build a lasting and stable peace regime" on the Korean Peninsula.

The White House has yet to release the document's text. But it was photographed by the news media during a signing ceremony.

The document lays out four broad commitments. It says the sides "commit to establish new U.S.-DPRK relations in accordance with the desire of the peoples of the two countries for peace and prosperity."

And it says they will commit to recovering the remains of prisoners of war and those missing in action.

It's unclear exactly what Trump has promised Kim in terms of security. The U.S. has committed to providing "security guarantees" to the North.

Secretary of State Mike Pompeo declined to say Monday whether guarantees might include withdrawing U.S. troops from the Korean Peninsula.

UPDATE #2:

President Donald Trump is praising North Korean leader Kim Jong Un as a "very worthy, very smart negotiator" on behalf of his people as the two leaders bid each other farewell after their historic summit.

Trump was asked by reporters in Singapore during his final appearance with Kim on Tuesday what surprised him most during their meetings.

Trump says Kim has a "great personality" and is "very smart. Good combination."

Trump also says he learned Kim is "a very talented man" and "loves his country very much."

He's wrapping up the summit by saying the two had "a terrific day" and "learned a lot about each other and about our countries."

He says he expects they'll meet again many times.

North Korean leader Kim Jong Un has left the small Singapore island that was the site of his meeting with President Donald Trump.

Kim's convoy was left Sentosa Island on Tuesday afternoon after he signed a document with the American president, who stayed behind at the hotel where the two leaders met.

UPDATE #1:

President Donald Trump and North Korean leader Kim Jong Un have signed what Trump says is a "very important" and "pretty comprehensive" document.

But Trump is refusing to tell reporters what the declaration says. The document is set to be handed out to reporters later.

Trump said Tuesday as the leaders wrapped up their historic summit in Singapore that he and Kim "have developed a very special bond" during their day together.

And he says, "Both sides are going to be impressed with the result."

Kim told reporters that "the world will see a major change," though it's unclear how.

The summit marked the first between a sitting U.S. president and a North Korean leader.

President Donald Trump says he "absolutely" would invite North Korean leader Kim Jong Un to the White House.

After Kim and Trump signed what Trump called a "pretty comprehensive" document, Trump was asked about a possible invitation. Trump said "absolutely, I would" invite Kim.

Before Tuesday's summit in Singapore, Trump had dangled the prospect of a White House visit for Kim.

After the lunch, Trump announced the signing ceremony but provided no details, saying only: "We're going to be announcing that in a couple of minutes."

At a meeting that could chart the course for historic peace or raise the specter of a growing nuclear threat, both leaders expressed optimism. Kim called the sit-down a "good prelude for peace" and Trump pledged that "working together we will get it taken care of."

Giving voice to the anticipation felt around the world, South Korean President Moon Jae-in said Tuesday he "hardly slept" before the summit. Moon and other officials watched the live broadcast of the summit before a South Korean Cabinet meeting in his presidential office.

The meeting was the first between a sitting U.S. president and a North Korean leader.

After meeting privately and with aides, Trump and Kim moved into the luncheon at a long flower-bedecked table. As they entered, Trump injected some levity to the day's extraordinary events, saying: "Getting a good picture everybody? So we look nice and handsome and thin? Perfect."

Then they dined on beef short rib confit along with sweet and sour crispy pork.

Critics of the summit leapt at the leaders' handshake and the moonlight stroll Kim took Monday night along the glittering Singapore waterfront, saying it was further evidence that Trump was helping legitimize Kim on the world stage as an equal of the U.S. president. Kim has been accused of horrific rights abuses against his people. During his stroll, crowds yelled out Kim's name and jostled to take pictures, and the North Korean leader posed for a selfie with Singapore officials.

Trump responded to such commentary on Twitter, saying: "The fact that I am having a meeting is a major loss for the U.S., say the haters & losers." But he added "our hostages" are back home and testing, research and launches have stopped.

Trump also tweeted: "Meetings between staffs and representatives are going well and quickly ... but in the end, that doesn't matter. We will all know soon whether or not a real deal, unlike those of the past, can happen!"

The summit capped a dizzying few days of foreign policy activity for Trump, who shocked U.S. allies over the weekend by using a meeting in Canada of the Group of Seven industrialized economies to alienate America's closest friends in the West. Lashing out over trade practices, Trump lobbed insults at his G-7 host, Canadian Prime Minister Justin Trudeau. Trump left that summit early and, as he flew to Singapore, tweeted that he was yanking the U.S. out of the group's traditional closing statement.

As for Singapore, the White House said Trump was leaving early because negotiations had moved "more quickly than expected" but gave no details. The president planned to stop in Guam and Hawaii on the way back to Washington.

The unfolding summit was a remarkable change in dynamics from less than a year ago, when Trump was threatening "fire and fury" against Kim, who in turn scorned the American president as a "mentally deranged U.S. dotard." Beyond the impact on both leaders' political fortunes, the summit could shape the fate of countless people — the citizens of impoverished North Korea, the tens of millions living in the shadow of the North's nuclear threat, and millions more worldwide.

Alluding to the North's concerns that giving up its nuclear weapons could surrender its primary deterrent to forced regime change, Secretary of State Mike Pompeo told reporters that the U.S. was prepared to take action to provide North Korea with "sufficient certainty" that denuclearization "is not something that ends badly for them."

He would not say whether that included the possibility of withdrawing U.S. troops from the Korean Peninsula, but said the context of the discussions was "radically different than ever before."

"I can only say this," Pompeo said. "We are prepared to take what will be security assurances that are different, unique, than America's been willing to provide previously."

The North has faced crippling diplomatic and economic sanctions as it has advanced development of its nuclear and ballistic missile programs. Pompeo held firm to Trump's position that sanctions will remain in place until North Korea denuclearizes — and said they would even increase if diplomatic discussions did not progress positively.

Experts believe the North is close to being able to target the entire U.S. mainland with its nuclear-armed missiles, and while there's deep skepticism that Kim will quickly give up those hard-won nukes, there's also some hope that diplomacy can replace the animosity between the U.S. and the North.